Weighted homogeneous singularities and rational homology disk smoothings

Bhupal, Mohan; Stipsicz, András I.

doi:10.1353/ajm.2011.0036

Cited by 21 publications

(40 citation statements)

References 28 publications

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“…Proof. Fix (a, d) ∈ {(1, d), (2, d), (4, 4), (5,5), (5,6), (d, 7)}. Order the lines in the configuration C 0 , C 1 , · · · , C d where C 0 intersects all other lines in double points.…”

Section: 52mentioning

confidence: 99%

“…The special case in the previous example generalizes to a family of dually positive symplectic plumbings of spheres that can be completely rationally blown down, given by the graphs in figure 17a. This is the only family of dually positive graphs which has a symplectic rational blowdown (of the entire configuration) due to the classifications in [6] and [30].…”

Section: 4mentioning

confidence: 99%

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Symplectic fillings of Seifert fibered spaces

Starkston

2014

Trans. Amer. Math. Soc.

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Abstract. We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S 2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

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Section: 52mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Symplectic fillings of Seifert fibered spaces

Starkston

2014

Trans. Amer. Math. Soc.

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Thanks to the recent work [22] and [1], one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the 3 classes whose resolution dual graph has central vertex with valency 4.

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confidence: 99%

On rational homology disk smoothings of valency 4 surface singularities

Wahl

2011

Geom. Topol.

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Thanks to recent work of Stipsicz, Szabó and the author [22] and of Bhupal and Stipsicz [1], one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, that is, one with Milnor number 0. They fall into several classes, the most interesting of which are the 3 classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for those classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in [22]; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class.We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H -shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz [1], is evidence for a general conjecture:Conjecture The only complex surface singularities admitting a QHD smoothing are the (known) weighted homogeneous examples. 14B07, 14J17, 32S30Introduction Suppose f W .C 3 ; 0/ ! .C; 0/ is an analytic map germ such that .V; 0/ Á .f 1 .0/; 0/ has an isolated singularity at the origin. Its Milnor fibre M is a "nearby fibre" of f intersected with a small ball about the origin: M D f 1 .ı/ \ B .0/ (0 < jıj j j). M is a manifold of dimension 4, compact with boundary the link L of the singularity, the intersection of X with a small sphere. L is a compact oriented 3-manifold, which can be reconstructed from any resolution of the singularity as the boundary of a tubular neighborhood of the exceptional set. M is simply-connected, and has the homotopy It is natural to consider smoothings of (germs of) arbitrary normal complex surface singularities. That is, one has a flat surjective map f W .V; 0/!.C; 0/, where .V; 0/ has an isolated 3-dimensional singularity, and the fibre .f 1 .0/; 0/Á.V; 0/ is a normal surface singularity. V has a link L as before, and Lê and Hamm showed how to define a Milnor fibre M -again, a Stein filling of L. M need no longer be simply connected, though b 1 .M /D0 (see Greuel and Steenbrink [6] (The W family is [24, 5.9.2].) We found other one-parameter infinite families; all were weighted homogeneous singularities, hence (see Orlik and Wagreich [17]) is star-shaped (a star is a vertex of valency at least 3). Most had a star of valency three; but there were also valency 4 families for each triple. We list the graphs below for each p 2, and name the type via a triple .a; b; cI d/:Geometry & Topology, Volume 15 (2011) Rational homology disk smoothingsThe main goal of the current paper is to give a new and uniform "quotient constr...

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“…As previously mentioned, (1) has been proved in [26]. The relevant expression in the Conjecture can be written in an alternative and suggestive way.…”

Section: Formulas For µ and τmentioning

confidence: 98%

Milnor and Tjurina numbers for smoothings of surface singularities

Wahl¹

2015

Alg. Geom.

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For an isolated hypersurface singularity {f = 0}, the Milnor number µ is greater than or equal to the Tjurina number τ (the dimension of the base of the semi-universal deformation), with equality if f is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity (V, 0), the difference µ − τ can be defined whether or not (V, 0) is smoothable; the author has proved it is non-negative, and equal to 0 if and only if (V, 0) is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface singularities. Here, µ − τ must be modified so that it is independent of any smoothing. This expression, involving cohomology of exterior powers of the bundle of logarithmic derivations on the minimal good resolution, is conjecturally non-negative, with equality iff (V, 0) is quasi-homogeneous. We prove the "if" part; identify special cases where the conjecture is particularly interesting; verify it in some non-trivial cases; and prove it for a Q-Gorenstein smoothing when the index one cover is a hypersurface. This conjecture arose regarding the classification of surface singularities with rational homology disk smoothings.

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Weighted homogeneous singularities and rational homology disk smoothings

Abstract: We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weight.

Cited by 21 publications

References 28 publications

Symplectic fillings of Seifert fibered spaces

Symplectic fillings of Seifert fibered spaces

On rational homology disk smoothings of valency 4 surface singularities

Milnor and Tjurina numbers for smoothings of surface singularities

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